You drop a rubber ball from a height of 48 inches. Each time it bounces, it reaches lower and lower heights. The following sequence shows its height with each successive bounce. How high will the ball bounce on its fifth bounce?

48, 36, 27, 20.25,...

Guidance

When looking at a sequence of numbers, consider the following possibilities.

There could be a
common difference
(the same value is added or subtracted) to progress from each term to the next.

Example:
(add 3)

There could be a
common ratio
(factor by which each term is multiplied) to progress from one term to the next.

Example:
,
multiply by

If the terms are fractions, perhaps there is a pattern in the numerator and a different pattern in the denominators.

Example:
(numerator (+2), denominator (-1))

If the terms are growing rapidly, perhaps the difference between the term values is increasing by some constant factor.

Example:
(add 3, add 4, add 5, ...)

The terms may represent a particular type of number such as prime numbers, perfect squares, cubes, etc.

Example:
(prime numbers)

Consider whether each term is the result of performing an operation on the two prior terms.

Example:
(add the previous two terms)

Consider the possibility that the value is connected to the term number:

Example:

In this example

This list is not intended to be a comprehensive list of all possible patterns that may be present in a sequence but they are a good place to start when looking for a pattern.

Example A

Find the next two terms in the sequence:

Solution:
Each term is the result of multiplying the previous term by
. Therefore, the next terms are:

and

Example B

Find the next two terms in the sequence:

Solution:
The difference between the first two terms
is 3, the difference between the second and third terms
is 4, the difference between the third and fourth terms
is 5 and the difference between the fourth and fifth terms
is 6. Each time we add one more to get the next term. The next difference will be 7, so
for the sixth term. To get the seventh term, we add 8, so
.

Example C

Find the next two terms in the sequence:

Solution:
This sequence requires that we look at the previous two terms. To get the third term, the second term was subtracted from the first:
. To get the fourth term, the third term is subtracted from the second:
. Similarly:
. Now, to get the next terms, continue the pattern:

and

Intro Problem Revisit
Each successive term in the sequence is the result of multiplying the previous term by
. Therefore, the next term, the fifth, is:

.

Therefore, the ball reaches a height of 15.1875 inches on its fifth bounce.

Guided Practice

Answers

1. Each term is the previous term plus 4. Therefore, the next two terms are 11 and 15.

2. The pattern here is somewhat hidden because some of the fractions have been reduced. If we “unreduced” the second and fourth terms we get the sequence:
,
,
,
. Now the pattern can be observed to be that the numerator and denominator each increase by 3. So the next two terms are
and
. Reducing the last term gives us the final answer of
and
.

3. This sequence is the set of perfect squares or the term number squared. Therefore the
and
terms will be
and
.

Vocabulary

Sequence

An arrangement of numbers which follows a pattern.

Common Difference

The constant value which is repeatedly added to each term in an arithmetic sequence to obtain the next term.

Common Ratio

The constant value which is multiplied by each term in a geometric sequence to obtain the next term.